5. (a) Show algebraically that() + (¹)(c)=+where n, r and z are positive integers with x

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Answered: 5. (a) Show algebraically that () where…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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14. Suppose a and b are single-digit positive integers chosen indepetndently and at random. What is the probability that the point (a, b) lies above the parabola y = ar? – br ? (A) (B) (C)을 (D) 11 13 5 17 19 (E) 81 81 27 81
7. Complete the spreadsheet section below so that columns A through C are complete for numbers 1 to 100. (The value for a should be a random number generated by the formula in cell E1.) A B C D E 1 X x^2 x^2+a a =RANDBETWEEN(-10,10) 2 1 2
2. a) Let X and Y be independent identically distributed binomial random variables, X~ Bin(3, 0.4) and Y ~ Bin(3, 0.4). i) Compute the probability P(X = 0). ii) Compute the probability that the product XY equals 0, that is, compute P(XY = 0). b) The actual amount of jam (in g) that a filling machine puts into "150 g" jars may be looked upon as a random variable having a normal distribu- tion with σ = 6 and mean μ = 155. What proportion of jars, in the long-run, contain less than 150 g? N(0, 1), Give your answer in terms of the function, where for Z (z) = P(Z), and a give a sketch of the normal curve with a shaded area that corresponds to the required probability.
8.) Six letters of the alphabet are chosen at random. Find the probability that the following "words" are formed: (a) REDSOX (b) A "word" that ends in Y (c) A "word" with no repeated letters (d) A "word" that does not contain A,B, or C
The complement of an event is all possible outcomes not in the event. For example: If A = at least 3 girls are born to a family with 6 kids. Then A' = A compliment is what happens if there are not at least 3 girls = less than 3 girls born to the family. A useful formula: P(A) + P(A' ) = 1 or P(A') = 1 - P(A). If P(A) = 0.194, what is P(A')?
Suppose that X is a random variable * ? having the following m.g.f., Find Var(x) 7t_e4t M, (t) = eT 173 O 5 O 413 O 3148 O 55 O
2. Suppose one has n non-degenerate random variables X1, X2,, X, so that X1+ X2 + · · · + Xn = L for some constant L. (Recall that a random variable is non-degenerate if it is not a constant in disguise.) Show that there must be at least one pair of indices i j so that p(X;, X;) < 0.
Q.3. Compute the probability that: (a) their sum is odd; (b) their product is even. Three distinct integers are chosen at random from the first 15 positive integers.
1. Given that X~N(39,32)A Random variable X is normally distributed and has a mean of 9 C Random variable X is not normally distributed and the mean is 39 B Random variable X is normally distributed and has a mean of 3 D Random variable X is normally distributed and the mean is 39
24. Gaussian random variables X, and X,, for which, X, = 2, o = 9, X, =-1, ox, = 4 and Cxx, =-3, are transformed to new random variables Y, and Y, according to जापर डेरें| Y, =-X, + X, Y, =-2X,- 3X, (iii) Cy,Cy, .. Find
4 Suppose that Alejandro is planting a garden of tulips. Let X be the Bernoulli random variable that returns 1 if the tulip is red, and 0 otherwise. Suppose Y is another Bernoulli random variable, independent of X, that returns 1 if the tulip survives longer than 30 days, and 0 otherwise. Both X and Y have parameters 1/2. Let Z be the random variable that returns the remainder of the division of X +Y by 2. For example, if X = 1 and Y = 0, Z = remainder() = 1 (a) Prove that Z is also a Bernoulli random variable, also with parameter 1/2. (b) Prove that X, Y, Z are pairwise independent but not mutually independent. (c) By computing Var[X+Y+Z] according to the alternative formula for variance and using the variance of Bernoulli r.v.'s, verify that Var[X+Y+Z] = Var[X]+Var[Y] +Var[Z] %3D (observe that this also follows from the proposition on slide 5 of the lecture segment entitled "Binomial distribution").
Show that the average time till the first head is observed is y₁= p-¹, where p is the probability of getting a head each time the coin is tossed. Then find a closed form explicit solution to yn, the average time till n consecutive heads are observed for the first time.

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5. (a) Show algebraically that () where n, r and z are positive integers with r < r and z